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The solution of a partial differential equation can be obtained by computing the inverse operator map between the input and the solution space. Towards this end, we introduce a \textit{multiwavelet-based neural operator learning scheme}…
In this work, we introduce the notion of Gevrey asymptotic expansion and we show how the classical concept of a convergent power series can be generalized to include the case in which the radius of convergence is zero. This technique can be…
We present a general formula for the particular solution of an inhomogeneous linear difference equation with variable coefficients. The answer is expressed as a weighted sum of fundamental solutions of the associated linear difference…
In this paper, we discuss some problems of elementary plane differential geometry and kinematics. Although the results are not new, the consistent use of complex-valued functions (plane curves) of a real variable (parameter) allows to…
The method of this paper is my original creation. A new method for solving linear differential equations is proposed in this paper. The important conclusion of this paper is that arbitrary order linear ordinary differential equations with…
Symmetry reductions of systems of two nonlinear partial differential equations are studied. We find ansatzes reducing system of partial differential equations to system of ordinary differential equations. The method is applied to system…
We introduce the concept of numerical Gaussian processes, which we define as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes,…
Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and…
We study generalized diffusion-wave equation in which the second order time derivative is replaced by integro-differential operator. It yields time fractional and distributed order time fractional diffusion-wave equations as particular…
In this paper, we analyze nonlinear differential equations subject to generalized boundary conditions. More specifically, we provide a framework from which we can provide conditions, which are straightforward to check, for the solvability…
This paper proposes a novel Generalized Non-Standard Finite Difference (GNSFD) scheme for the numerical solution of a class of fractional partial differential equations (FrPDEs). The formulation of the method is grounded in optimization and…
In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [10] for convection-diffusion equations, which relies on a…
A new boundary value problem for partial differential equations is discussed. We consider an arbitrary solution of an elliptic or parabolic equation in a given domain and no boundary conditions are assumed. We study which restrictions the…
A large class of initial-boundary value problems of linear evolution partial differential equations formulated on the half-line is analyzed via the unified transform method. In particular, explicit formulae are presented for the generalized…
We study generalized additive partial linear models, proposing the use of polynomial spline smoothing for estimation of nonparametric functions, and deriving quasi-likelihood based estimators for the linear parameters. We establish…
We give a lower bound for the Gaussian curvature of convex level sets of minimal graphs and the solutions to semilinear elliptic equations with the norm of boundary gradient and the Gaussian curvature of the boundary.
The asymptotic behavior of the analytic solutions of a family of singularly perturbed q-difference-differential equations in the complex domain is studied. Different asymptotic expansions with respect to the perturbation parameter and to…
Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to…
These lecture notes for a graduate course present an introduction to the mathematical theory of finite element methods for the numerical solution of partial differential equations. Covered are conforming and nonconforming (in particular,…
In this present paper, we introduce and study a dynamical systems involving fractional derivative operator and nonlocal condition, which is constituted of a fractional evolution equation and a time-dependent variational inequality, and is…